On the surface average for harmonic functions: a stability inequality
In this article we present some of the main aspects and the most recent results related to the following question: If the surface mean integral of every harmonic function on the boundary of an open set D is "almost'' equal to the value of these functions at x0 in D, then is D "...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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University of Bologna
2024-01-01
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| Series: | Bruno Pini Mathematical Analysis Seminar |
| Subjects: | |
| Online Access: | https://mathematicalanalysis.unibo.it/article/view/18860 |
| _version_ | 1797359627526995968 |
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| author | Giovanni Cupini Ermanno Lanconelli |
| author_facet | Giovanni Cupini Ermanno Lanconelli |
| author_sort | Giovanni Cupini |
| collection | DOAJ |
| description | In this article we present some of the main aspects and the most recent results related to the following question: If the surface mean integral of every harmonic function on the boundary of an open set D is "almost'' equal to the value of these functions at x0 in D, then is D "almost'' a ball with center x0? This is the stability counterpart of the rigidity question (the statement above, without the two "almost'') for which several positive answers are known in literature. A positive answer to the stability problem has been given in a paper by Preiss and Toro, by assuming a condition that turns out to be sufficient for ∂D to be geometrically close to a sphere. This condition, however, is not necessary, even for small Lipschitz perturbations of smooth domains, as shown in our recent paper, in which a stability inequality is obtained by assuming only a local regularity property of the boundary of D in at least one of its points closest to x0. |
| first_indexed | 2024-03-08T15:26:29Z |
| format | Article |
| id | doaj.art-672233b9ab1741e9ab52eb4fb72388b8 |
| institution | Directory Open Access Journal |
| issn | 2240-2829 |
| language | English |
| last_indexed | 2024-03-08T15:26:29Z |
| publishDate | 2024-01-01 |
| publisher | University of Bologna |
| record_format | Article |
| series | Bruno Pini Mathematical Analysis Seminar |
| spelling | doaj.art-672233b9ab1741e9ab52eb4fb72388b82024-01-10T08:48:10ZengUniversity of BolognaBruno Pini Mathematical Analysis Seminar2240-28292024-01-0114212913810.6092/issn.2240-2829/1886017222On the surface average for harmonic functions: a stability inequalityGiovanni Cupini0Ermanno Lanconelli1Dipartimento di Matematica, Università di BolognaDipartimento di Matematica, Università di BolognaIn this article we present some of the main aspects and the most recent results related to the following question: If the surface mean integral of every harmonic function on the boundary of an open set D is "almost'' equal to the value of these functions at x0 in D, then is D "almost'' a ball with center x0? This is the stability counterpart of the rigidity question (the statement above, without the two "almost'') for which several positive answers are known in literature. A positive answer to the stability problem has been given in a paper by Preiss and Toro, by assuming a condition that turns out to be sufficient for ∂D to be geometrically close to a sphere. This condition, however, is not necessary, even for small Lipschitz perturbations of smooth domains, as shown in our recent paper, in which a stability inequality is obtained by assuming only a local regularity property of the boundary of D in at least one of its points closest to x0.https://mathematicalanalysis.unibo.it/article/view/18860surface gauss mean value formulastabilityharmonic functionsrigidity |
| spellingShingle | Giovanni Cupini Ermanno Lanconelli On the surface average for harmonic functions: a stability inequality Bruno Pini Mathematical Analysis Seminar surface gauss mean value formula stability harmonic functions rigidity |
| title | On the surface average for harmonic functions: a stability inequality |
| title_full | On the surface average for harmonic functions: a stability inequality |
| title_fullStr | On the surface average for harmonic functions: a stability inequality |
| title_full_unstemmed | On the surface average for harmonic functions: a stability inequality |
| title_short | On the surface average for harmonic functions: a stability inequality |
| title_sort | on the surface average for harmonic functions a stability inequality |
| topic | surface gauss mean value formula stability harmonic functions rigidity |
| url | https://mathematicalanalysis.unibo.it/article/view/18860 |
| work_keys_str_mv | AT giovannicupini onthesurfaceaverageforharmonicfunctionsastabilityinequality AT ermannolanconelli onthesurfaceaverageforharmonicfunctionsastabilityinequality |